Yuan-Chuan Li

On generators of integratedC-semigroups andC-cosine functions

The following two theorems are proved: (1) the generator of an exponentially equicontinuousn-times integratedC-cosine function also generates an exponentially equicontinuous [(n+1)/2]-times integratedC-semigroup; (2) IfA and −A are generators of exponentially equicontinuousn-times integratedC-semigroups, thenA2 generates an exponentially equicontinuousn-times integratedC-cosine function.

On Local α-Times Integrated C-Semigroups

This paper presents several characterizations of a local α-times integrated C-semigroup {T(t);0≤tlτ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem. We also discuss properties concerning the nondegeneracy of T(⋅), the injectivity of C, the closability of subgenerators, the commutativity of T(⋅), and extension of solutions of the associated abstract Cauchy problem.

Integrated C-Semigroups and C-Cosine Functions of Hermitian and Positive Operators

Some recent results on n-times integrated C-semigroups and C-cosine functions of hermitian and positive operators on Banach spaces are discussed. The following are some interesting properties: 1) A hermitian n-times integrated C-semigroup T() (resp. C-cosine function C()) is (n — 1)-th continuously differentiable in operator norm on [0, ∞) and T( n-1 ) (•) (resp. C (n-1)()) is a norm continuous hermitian once integrated C-semigroup (resp. C-cosine function) if n ≥ 2; 2) A hermitian C-semigroup is infinitely differentiable in operator norm on (0, ∞); 3) A hermitian C-cosine function is norm continuous either at all points or at none point of [0, ∞); 4) A positive C-semigroup (resp. C-cosine function) which dominates C is infinitely differentiable in operator norm on [0, ∞); moreover, if it is nondegenerate, then its generator A must be bounded and \(T(t) = \sum\nolimits_{n = 0}^\infty {\frac{{{t^n}}}{{n!}}} {A^n}C (resp. C(t) = \sum\nolimits_{n = 0}^\infty {\frac{{{t^{2n}}}}{{(2n)!}}{A^n}C);} 5) \) Hermitian n-times integrated semigroups and hermitian n-times integrated cosine functions are exponentially bounded.

Hermitian and Positive Integrated C-cosine Functions on Banach Spaces

Peculiar properties of hermitian and positive n-times integrated C-cosine functions on Banach spaces are investigated. Among them are: (1) Any nondegenerate positiven -times integrated C-cosine function is infinitely differentiable in operator norm; (2) An exponentially bounded, nondegenerateC -cosine function on Lp(μ) (1<p<∞) (orL1(μ), C0 , in case C has dense range) is positive if and only if its generator is bounded, positive, and commutes with C.

An abstract ergodic theorem and some inequalities for operators on Banach spaces

We prove an abstract mean ergodic theorem and use it to show that if {An} is a sequence of commuting m-dissipative (or normal) operators on a Banach space X, then the intersection of their null spaces is orthogonal to the linear span of their ranges. It is also proved that the inequality IIx+AyII > II x -2 \/IAxII IIyHI (x, y G D(A)) holds for any m-dissipative operator A. These results either generalize or improve the corresponding results of Shaw, Mattila, and Crabb and Sinclair, respectively.

Some characterizations of convex functions

The main result in this paper is to establish some new characterizations of convex functions, in which we also simplify the proof of the characterizations given by Bessenyei and Pales.

Contractibility of compact contractions in Hilbert space

For a finite set Σ of compact contractions in a complex Hilbert space (H·∥·∥), it is shown that r(A)<1 for all A in the multiplicative semigroup generated by Σ if and only if there exists a positive integer N such that ∥A∥<1 for all A in the multiplicative semigroup generated by Σ with length greater than N. Here r(A) denotes the spectral radius of A. As an application, an answer is given to an infinite-dimensional case of the finiteness conjecture for the generalized spectral radius attributed to J.C. Lagarias and Y. Wang [Linear Algebra Appl. 214 (1995) 17].